A third and fully automated way (that can also be used for LTL model checking) is to translate the negation of the formula to an equivalent $\omega$-automaton and check its emptiness. For $GFa \to Fa$, we first abbreviate $Fa$ by $q_0$ and then $Gq_0$ by $q_1$ so that we finally obtain the following Büchi automaton for the negation $\neg(GFa \to Fa)$:

Looking at the state transition diagram reveals that the single initial state has no outgoing transition, so the $\omega$-automaton does not accept any word. Hence, the negation of the formula is unsatisfiable, and thus, the original formula is valid.